$$ \lim_{x \to 0} \frac{\lvert2x-1\rvert – \lvert2x+1\rvert}{x} $$
Defining the function piecewise reveals the limit is in fact, continuous about 0
However when I go to solve it in a normal algebraic manner, the $2x$ terms are canceling, leaving me with an undefined output – 0 in the denominator.
Any hints would be fantastic, I've solved this every way I can think of and I keep getting different answers, none of which are the correct answer.
I did graph this, and it does show a continuous function around $0$ where $f(x) = -4$
My problem here is that I'm being a huge dunce about absolute values.
Best Answer
For $\;x\;$ pretty close to zero, $\;2x-1<0\;,\;\;2x+1>0\;$ , so we have the limit
$$\frac{-2x+1-2x-1}x=\frac{-4x}x=-4\xrightarrow[x\to 0]{}-4$$